In this paper, we study the problem of non parametric estimation
of the stationary marginal density f of an α or a
β-mixing process, observed either in continuous time or in
discrete time. We present an unified framework allowing to deal
with many different cases. We consider a collection of finite
dimensional linear regular spaces. We estimate f using a
projection estimator built on a data driven selected linear space
among the collection. This data driven choice is performed via the
minimization of a penalized contrast. We state non asymptotic risk
bounds, regarding to the integrated quadratic risk, for our
estimators, in both cases of mixing. We show that they are
adaptive in the minimax sense over a large class of Besov balls.
In discrete time, we also provide a result for model selection
among an exponentially large collection of models (non regular
case).

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